Friday, 23 August 2013

Oedipus and the difficult relationship between maths and economics

Noah Smith recently wrote a “rant”
on maths in (macro)economics, which elicited a swift response from
Paul Krugman, which in turn prompted Bryan Caplan to argue that
“Economath fails the cost benefit analysis”. As a mathematician
(i.e. someone who is employed by a maths department of a university)
I think all the commentators make valuable points but, maybe somewhat
surprisingly, my strongest affinity is with Bryan Caplan’s position
that maths is failing economics. What I intend to do in this post is
present an argument that, historically, `physics maths' has been driven
by economic intuition and the contemporary problems in economics are
that it is adopting maths from the physical science rather than
generating a more insightful mathematics of its own. In the context
of Noah’s original article, econmath is not enjoyable in the same
way that ill fitting clothes are not enjoyable, or playing football
in tennis shoes can be painful Metaphorically, maths is Oedipus and
economics is King Laius: maths does not recognise its father, father
does not recognise child, and tragedy follows.

In a recent post Caplan challenges Krugman to “Name three important
economic insights you think we owe to economath.”, I will support
my case by naming three critical developments in “physics
math” that come out of economic intuition.

The
mathematisation of physics

Aristotle, and
his successors such as the medieval Islamic scientists, did not think
that maths had anything to offer physics, probably similar to
Caplan’s view that maths has little to offer economics. Maths had
nothing to say about causes, what was important in understanding
nature were the qualities of objects, whether they were heavy or
light, hot or cold, wet or dry, hard or soft. Since mathematics is
concerned with precision, numbers and proportions, it could not give
any insight into these qualities; “boulders were never perfectly
spherical nor were pyramids perfectly pyramidal, so what was the use
of treating them as such” [7,
p 16]. The critical distinction for Greek science and its descendants
was that mathematicians dealt with pure abstract objects while
natural philosophers and engineers work with the concrete, messy,
qualities of physical objects [9,
p 65]. This approach was not just Greek, other cultures, such as the
Chinese, took a similar line on the usefulness of maths to science
[11,
p 53].

Approaching physics through mathematics was uniquely European and was
first recorded by the ‘Merton Calculator’ Thomas Bradwardine in

[Mathematics]
is the revealer of genuine truth, for it knows every hidden secret
and bears the key to every subtlety of letters. Whoever, then, has
the effrontery to pursue physics while neglecting mathematics should
know from the start that he will never make his entry through the
portals of wisdom [13,
p 176]

Bradwardine had
entered Merton College in 1323 and twelve years later went on work
for the Bishop of Durham, who was the Treasurer and Chancellor of
England, was eventually appointed Archbishop of Canterbury shortly
before his death in 1349.

Arguing that physics was inseparable from mathematics was
revolutionary because it completely broke with Aristotle’s approach
to physics. From this point ‘Latin’ science would take a very
different path to Islamic science, which was based on the same
Hellenistic tradition. Bradwardine put this theory into practice by
looking for a mathematical description of Aristotle’s laws of
motions in his 1328 book De proportione velocitatum in motibus
(‘Concerning the ratio of speeds in movement’), he achieved
this, but being based on Aristotle’s incorrect principles, it was
wrong. The ideas were developed by subsequent Calculators and the
French philosophers Jean Buridan, who identified the concept of
inertia, and Nicolas Oresme, who identified the mean speed theorem.

Key to unlocking the laws of motion was the realisation that speed
was the ratio of distance divided by time. This seems obvious today
but was not to Aristotle who had investigated measurement in Physics
and Metaphysics, parts of the Organon where he
claimed that a measure is always the minimum unit of the thing being
measured. The measure shares the same substance as the subject of
measurement. For example, numbers are measured by the smallest
number, ‘1’, distances are measured by the smallest length of
distance, say an inch, while liquids are measure in pints and so
forth. This is not conceptually unnatural, the Chinese language still
incorporates ‘measure words’ when quantifying objects. When
Aristotle measured physical objects, as Richard Hadden has noted, “he
[was] careful not to mix quantities differing in kind in the same
expression” [12,
p 75], he, and scientists in the Greek tradition could not get their
heads around the very concept of ‘metres per second’ or
‘foot—pound’.

So why did Bradwardine ‘get it’? Joel Kaye has argued that the 45
Fellows at Merton were not just occupied in academic activities but
were involved in the daily management of the College’s extensive
resources

The
three bursars who had collective responsibility for the college’s
monies, the warden who headed the yearly audit and visited the
far-flung manors at harvest time to assess the year’s taxation, in
money or in kind, [others] oversaw the books and calculated the
profits of the college’s many properties [17,
p 33]

The point is
Bradwardine would have developed his mathematical skills through
economic application. But this is insignificant on its own, what was
critical was the change in the conception of measurement that had
occurred when Albert the Great studied Aristotle’s Nicomachean
Ethics, in the 1250s after it had been translated into Latin in
1248/9.

Book V of Nicomachean Ethics considers the justice of economic
exchange, and as a philosophical text it is disjointed and hard to
follow. However, what is clear is that Aristotle sees reciprocal,
fair, exchange in the market as being fundamental to a well
functioning society since it binds individuals together [17,
p 51]. Since Nicomachean Ethics was concerned with justice,
Aristotle needed to identify under what conditions economic exchange
was ‘just’ and he did this by insisting that there needed to be
an equality between the goods being exchanged [17,
p 45]. Aristotle identified that for equality to be established there
needed to be a measure of the value of the goods, and this measure,
the price, was provided by money.

all
things that are exchanged must somehow be comparable. It is for this
end that money has been introduced, and it becomes in a sense an
intermediate; for it measures all things, and therefore the excess
and the defect — how many shoes are equal to a house or to a given
amount of food [17,
p 47 quoting Ethics]

In his
translation, Grosseteste described money as the medium of
exchange. Today we interpret this in the sense that money is a
physical token, however this is a modern interpretation. For the
medieval scholar the Latin word medium was more commonly used
in the sense of a mediator or intermediary, money is a
neutral agent that links two distinctive commodities, such as shoes
and houses.

Albert realised that if Aristotle was right about money being a
measure he could not be right about a measure sharing the substance
of the measured. This insight enabled Albert, and his successors, to
revolutionise the concept of measurement, in a way that contemporary
Muslim scholars did not. In particular students of Albert were able
to reject Aristotle’s theories of measurement and consider concepts
like metres per second or kilograms—metres per second (momentum).
[17,
p 67]

The mathematisation of Western Science is a consequence of the
ethical assessment of an en economic activity.

The
development of calculus

Bertrand Russell
(amongst others) stresses the fundamental role probability (and
statistics) has in science [21,
page 301], it is less well known that probability emerges out of the
ethical assessment of commercial contracts following Aristotelian
concepts of Justice. Even less well known is the role financial
practice and theory had in the genesis of Newton’s calculus on
which his physics is based.

Many people appreciate that Arabic (Hindu) numbers were introduced
into Europe through Fibonacci’s Liber Abaci, a financial
text book. But Fibonacci employed fractions, not decimals. The
significance of the decimal notation was highlighted by the Flemish
mathematician Simon Stevin in his 1585 text , De Thiende (‘The
tenths’). Stevin had not invented decimal fractions, they had been
used by the Arabs and Chinese and first appear in Europe in a German
text on algebra of 1525, but the audience for De Thiende was
‘practical men’ and Stevin pointed out that ‘all computations
that are met in business may be performed’ using his notation.
Stevin’s notation was in fact a bit cumbersome, decimal fractions
as we know them appear in English in 1616. [2,
p 316—317] Its also worth pointing out that the ‘=’ sign was
introduced in The Whetstone of Witte written by Robert Recorde
who controlled the English Mint).

The significance of decimal fractions to calculus is in how Newton
tackled the problem of motion, building on Oresme’s work. While in
Lincolnshire avoiding the 1666 Plague, Newton thought about how a
point turned into a line by moving in an infinitesimal moment, for
example how a pencil-line is drawn on a piece of paper. Newton called
the resulting curve a fluent and he called the velocity of how
the fluent grew in a moment its fluxion. For example a fluent
could be the distance of a cannon ball from a cannon, the fluxion its
velocity, or a force (the product of mass and acceleration) on cannon
ball was the fluxion and its momentum (the product of mass and
velocity) the fluent. Despite coming up with these ideas before 1667,
Newton only circulated them in 1671 in Tractatus demethodis
serierum et fluxionum (‘A Treatise on the Methods of Series and
Fluxions’) [16,
p 462].

Following Descartes’ Newton was comfortable in describing the
distance travelled by an object with a function, and following
Buridan and Oresme he understood the fluxion was the rate of change
of the fluent, but in order to solve problems in general he needed to
develop a way of deriving a function for a fluxion from a function
for a fluent.

The key to unlocking this puzzle lay in decimal notation. before
Stevin, numbers were generally written as composed or continued
fractions. For example, as a composed fraction the sum of money five
pounds, seven shillings and nine pence £5 and 149 pence, or 5would be written as a composed fraction,

as a continued
fraction,

while in decimal
notation it is

Newton realised this
was the same as writing

and was interested in
whether a similar approach could be taken with functions.

Newton’s approach, to write a function as a series

was not unheard of.
Independently Indian mathematicians had considered writing functions
as power series as early as the fourteenth century while in 1688
Nicolas Mercator considered power series for logarithms. (Mercator
was German and originally called Niklaus Kauffman, he Latinised his
name to Mercator; both Mercator and Kaufmann mean ‘merchant’). If
a decimal number, written as a continued fraction, never ends it is
an ‘irrational’ number rather than a ‘rational’ one. If a
function when written as a power series/polynomial has a finite
number of terms it is ‘algebraic’, if it has an infinite number
of terms it transcends algebraic functions, it is transcendental, and
usually given a name, such as sin, log, exp, etc.

Having realised that he could write a function as a power-series,
working out how the function changed could be resolved if he could
work out how a simple power changes, by establishing

What follows are two
supreme achievements of Western science, Newtonian mechanics and
Mathematical Analysis.

At this point it is worth observing that the diagrams in Noah’s
original blog post are both directly related to these first two
examples.

In
the early 1950s I was able to locate by chance this unknown book by a
French graduate student in 1900 rotting in the library of the
University of Paris and when I opened it up it was as if a whole new
world was laid out before me. [4,
13:00]

The book
Samuelson refers to is Louis Bachelier’s thesis Théorie de la
spéculation. The story that Samuelson tells, and has been
disseminated, associates Bachelier’s work with Einstein’s work on
Brownian motion rather than the pre-existing theories around finance.
The alignment of Bachelier’s work with theoretical physics, rather
than highlighting how the practice of applying mathematics to
economics informs the development of mathematics, is an example of
the tendency of academics to elevate theory over practice [10,
Chapter IV].

Bachelier’s work had nothing to do with the physical process of
Brownian motion, which Einstein was interested in, but was part of a
long (French) tradition of employing the Binomial random walk model
in finance. The canonical origins of mathematical probability are in
the Pascal-Fermat solution to the Problem of Points, which introduced
the model in 1654, a decade before calculus appears. Vernacular
research into the subject includes French actuary Emmanuel-Etienne
Duvillard’s Rechererches sur les rentes, les emprunts et les
remboursements (‘Researches on annuities, loans and refunding’)
[23]
and Jules Regnault’s Calcul des Chances et Philosophie de laBourse (‘Probability and the science of the markets’) of
1863 [15,
.] In 1870, Henri Lefèvre de Chateaudun, an actuary who had been the
private secretary of Baron de Rothschild [14]
published Traité des valeurs mobilières et des opérations de
Bourse: Placement etspéculation (‘Treatise of
financial securities and stock exchange operations’)” [20].
In 1875 the Danish actuary and astronomer Thorvald Thiele introduced
the concept of the random walk to model observational errors,
incorporating a dynamic component to Gauss’ earlier work [18].
In 1894 Poincaré, France’s greatest mathematician of the time,
chose to teach probability over all other subjects. At the time it
was traditional for lecture courses by prominent mathematicians to be
edited and published. In the case of Poincaré’s Calcul desProbabilitités, the editing of the lectures was carried out
Albert Quiquet, an actuary working for the La Nationale insurance
company, and not a university student, as was usual [3,
p 279], highlighting the active interest of practising financiers in
theoretical probability.

It was into this tradition that Bachelier was drawn. On account of
his parents’ early deaths, Bachelier had been unable to enter
university after school, and had worked initially for the family’s
firm in Normandy and then moved to Paris where he traded rentes
(i.e. consols) while studying mathematics at the University of
Sorbonne, eventually taking his doctorate nominally under Poincaré
([6],
[22]).
Bachelier’s thesis is known for extending the discrete time
Binomial random walk model into a continuous time model (i.e. it is a
descendant of Newton’s work on calculus), and this is the aspect
Samuelson picked up on.

Bachelier was not a succesful academic. It is rather obscure how
Bachelier earned his living in the following decade, he obtained a
few scholarships and in 1909 became a ‘free’ (unpaid) lecturer at
the Sorbonne, lecturing on probability theory applied to finance. [6]
In 1912 he published Calcul desProbabilitiés
(‘Probability Calculus’) and then in 1914, Le Jeu, la
Chance et le Hasard (‘Game, Chance and Randomness’). That
same year, on the verge of being permanently appointed to the
University of Paris, he, along with every other fit young Frenchman,
was conscripted, as a private, into the army. He survived the
cataclysm of the war, finishing it as a lieutenant, and in 1919 he
took up a temporary post at the University of Besançon. He had
further temporary positions at Dijon, between 1922 and 1925 and
Rennes between 1925 and 1927.

In 1926 a permanent position had become available at Dijon, which
Bachelier applied for. His application was reviewed by a professor at
Dijon who was not familiar with Bachelier’s work but believed an
important article published in 1913 contained a ‘gross error’.
The referee wrote to a young ‘doctoral brother’ (student of the
same doctoral adviser) who was developing a reputation in
probability, Paul Pierre Lévy, to comment on Bachelier’s work.
Unlike Bachelier’s unconventional pathe into academia, Lévy was
the son of an academic at the École Polytechnique, where he
studied and published his first paper at the age of 19 in 1905
becoming a professor at the École Nationale Supérieure des Mines
in 1913, and spent the war doing research for the French
artillery. In 1920 he was appointed to the École Polytechnique
and it was on the basis of his 1925 book Calcul des
Probabilitiés that he was asked to report on Bachelier.

Lévy checked the page that contained the suspected error and agreed
with the referee, Bachelier was blackballed. The issue that the
referee and Lévy had with the paper was that it appeared to
contradict a feature of Weiner’s 1921 formulation of Brownian
Motion, the fact was it didn’t if you followed the approach
Bachelier had been taking since his dissertation. The reviewer’s
were unfamiliar with this work and so the 1913 paper was ambiguous
and appeared wrong. Though the blackballing was painful to Bachelier,
in 1927, at the age of 57, he finally secured a permanent position at
Besançon where he would remain until his retirement in 1937. He
wrote two more books on probability in retirement, and died in 1946
in Brittany.

While Bachelier was never appointed to one of the great French
universities, perhaps because his vocational background did not
conform to France’s Rational ideal, he had a successful academic
career and to suggest his thesis was ‘rotting’ in a library
ignores the significant contribution Bachelier made to mathematics.
Notably his idea that probability was dynamic which was a significant
conceptual leap at the time and was taken up by Kolmogorov when
laying the foundations of modern probability.

In his thesis Bachelier discusses what he calls Rayonnement de la
probibilité (‘Radiation of probability’) [1,
p 46—47], [8,
p 41], the idea that a probability density evolves in time. This was
revolutionary, up until Bachelier had made his observation in his
thesis it had been assumed that probabilities were static, even
Bayesian approaches assumed the probability density was static but
you learnt more about it in time. The idea emerged in physics
following Einstein’s work in 1913 as the Fokker-Planck equation.

Kolmogorov became familiar with Bachelier’s work, and when Lévy
this he realised also that if he had looked into the Bachelier’s
work in 1927 when he refereed his application to Dijon, he might have
come to Kolmogorov’s conclusions before the Russian and be famous
today for laying the foundations of modern probability. Lévy
apologised to Bachelier and publicly acknowledged Bachelier’s
priority, not just over Wiener in the mathematical study of Brownian
motion, but also over over Kolmogorov in linking Brownian motion to
Fourier’s heat equation, and over himself in establishing certain
properties of the Wiener process. [22,
pp 20—21]

I think the Bachelier myth is incredibly important in demonstrating
how academics, from Lévy in 1927 to Samuelson in 1999, denigrate the
significance of economic phenomena in generating mathematical ideas
that have profound impact in the physical sciences.

Conclusions

I sympathise
with Bryan Caplan’s claim that modern mathematics obscures economic
intuition. I think Noah Smith’s point that econmath is unappealing
is because the mathematics developed in response to physical problems
is not suited to economic problems.

The observation that economists could do better with mathematics is
nothing new, the mathematician Augustin Cournot did not think that
economics was susceptible to precise quantification, in fact he was
wary of attempts to ‘arithmetise’ economics

There
are authors, like [Adam] Smith and Say, who, in writing on Political
Economy, have preserved all the beauties of a purely literary style;
but there are others, like [David] Ricardo, who, when treating the
most abstract questions, or when seeking great accuracy, have not
been able to avoid algebra, and have only disguised it under
arithmetical calculations of tiresome length. Any one who understands
algebraic notation, reads at a glance in an equation results reached
arithmetically only with great labour and pains.

I
propose to show in this essay that the solution of the general
questions which arise from the theory of wealth, depends essentially
not on elementary algebra, but on that branch of analysis which
comprises arbitrary functions, which are merely restricted to
satisfying certain conditions. [5,
p 4—5]

More recently
John von Neumann refused to write a review for Samuelson’s
Foundations of EconomicAnalysis in 1947 because “one
would think the book about contemporary with Newton”, like many
mathematicians who look at economics, on Neumann believed economics
needed better maths than it was being offered [19,
p 134].

Paul Krugman is right in emphasising the usefulness of mathematics as
a rhetorical device, the problem is that he does not recognise that
it is difficult to describe the Spitsbergen in winter in Arabic or
the pleasures of a desert oasis in Sami.

My plea is that economists stop using existing mathematics and start
commissioning new mathematics. I can see no resolution of the problem
of econmath until there is the sort of relationship between economics
and mathematics that mathematicians have with physical scientists and
mathematicians work with economists to create a mathematics that
enables the clear discussion of economic intuition.

In the Oedipus myth, although Oedipus does not realise he has killed his father, the murderer of Laius has to be brought to justice to end famine and pestilence in the kingdom. Ultimately it is mathematics that needs to correct the wrongs to economics.

Economists do not necessarily need new mathematics, they just need the right mathematics; dynamics and statistical mechanics.

Engineers and biologists (as well as physicists) have lots of dynamic models that could sensibly be moved across to economics. At the moment the the post-Godley Stock-Flow-Consistent research strand, along with Steve Keen, are the only economists who use a genuinely dynamic approach. If every ungraduate economist was forced to read Strogatz's 'Non-linear Dynamics & Chaos' macro would be solved within a decade.

As the example above of Bachelier, a statistical mechanical approach has been used in finance for many years. The Black-Scholes equation is after all just the diffusion equation; and by usage certainly the most successful equation in the history of economics/finance. Econophysicts have produced much useful work in finance on the basis of stat-mech.

Statistical mechanics is the obvious framework for many-bodied systems such as economics, yet despite its use in finance stat-mech has been resolutely ignored by economists.

Ian Wright's 'Social Architecture of Capitalism' uses simple stat-mech models that produce dozens of the regularaties that are seen in real economies. Despite this his papers have been resolutely ignored. Pascal Seppecher has independently produced ABM models that produce many of the same results as Wright's. My own stat-mech based work has mirrored the work of these two.

For an in depth expansion on the failed relationship between maths and economics read Mirowski's 'More heat than light'. The IS-LM diagrams, used for example by Nobel winning Krugman, are directly descended from field theory physics, a mathematical structure absolutely inappropriate for dynamic many-body systems such as economics and finance.

In a large sense, the question is: should economics attempt to be deterministic, as proper (classical) physics is, or to be non-deterministic? Each approach implies its maths. Making explicit my biases, physics should be deterministic via a proper geometry; economics should NOT attempt determinism beyond toy models. But, this should be a debate in both sciences as mainstream physics is now largely non-deterministic and mainstream economics is largely deterministic.

"The critical distinction for Greek science and its descendants was that mathematicians dealt with pure abstract objects while natural philosophers and engineers work with the concrete, messy, qualities of physical objects [9, p 65]."

This is pure drivel of a generalization http://en.wikipedia.org/wiki/Aristarchus_of_Samos

Aristarchus_of_Samos first defined the heliocentric system and made measurements of the size of the sun and distance to the moon.

Eudoxus is another example of someone who developed mathematics to deal with planetary motions. http://en.wikipedia.org/wiki/Eudoxus_of_Cnidus#Eudoxan_planetary_models

Apollonious was the inventor of conic sections Newton later used without giving any credit to describe planetary motions http://en.wikipedia.org/wiki/Apollonius_of_Perga

All of the above math and much more intended to be used to “save the phenomena”. equating all ancient Greek science and phisolophy with Aristotle is a straw man.

It is true that Aristotle said was that motion should cease when the cause is removed. Some interpreted that to mean that he said F = mv, as opposed F = ma, the Newtonian consequent (but not so clear how it arises from the Principia).

However, it is quite puzzling to anyone with a little brain that motion can continue when there is no application of force. Of course you could, as Galileo and Newton later did, define a universe where motion continues at constant speed when the force is zero. But that is purely axiomatic. There is no way to prove this experimentally and in any way Aristotle would like during his time. To understand why, just try to find any place in the universe where there are no forces. How do you know that forces are not present? Since we can only measure forces from their effects, can we really say that motion at constant speed is the effect of a causeless, zero-force- condition?

Now, this puzzled Aristotle. Of course, people with a shallow understanding of physics and philosophy are driven to easy conclusions such that Aristotle was wrong, blah, blah, blah... Try to understand that our physics nowadays are essentially metaphysical commitments that agree with some limited experimental observations. Quantum mechanics and general relativity, the two most accepted theories are two examples. There are at least 12 interpretations of quantum mechanics. In some locality is violated, in others counterfactual definitiveness is violated. Also all of our experiments are local in spacetime. This is why the strongest principle throughput the ages has been Pessimistic Meta-induction, i.e. that sooner or later all theories will be trashed.

Philosophy of science is a very complicated subject and not for easy conclusions.

As one trained in physics, I would like to comment on this most interesting history of math and economics, and the difficulties which economists have with the understanding of the proper application of math in the field. I think that what is important to keep in mind about physics is that its goal is to understand the inanimate world, and that this is an endeavor driven by the curiosity of the physicist. This curiosity led physicists ( by whom I mean anyone with an interest who does work investigating the inanimate world) to make observations about the world and to notice regularities in the world. Only then did attempts to apply math appear reasonable. The progress of physics may have evolved without the use of mathematics. That is, there is no a priori reason to believe that mathematics is related to the effort to understand the inanimate world. The fact that mathematics has played such an important role in physics is quite a marvel to physicists. Eugene Wigner gave a speech which summed up the relationship of physics and math quite nicely, which I highly encourage others to read. You can find it here: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html I think the title captures the ideas nicely: "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". In other words, math is used in physics not because physicist want to use math, but because it is so dramatically effective in describing the laws of nature.

Physics, at its root, is based on a belief system. That is, physicists believe that the universe is governed by immutable physical laws, and that we can discover these laws by making systematic observations of our world. That human beings can divine these laws through these methods. The language of mathematics has been found to be exceptionally effective in describing these natural laws. But not all of mathematics is applicable to physics, rather the physicist picks and chooses carefully what works from what is irrelevant. Wigner makes a key point that most mathematics used in physics was derived by the physicist as a necessary part of the investigation. Only later is it often realized that a description of the math was previously made by a mathematician. Efforts to relate and manipulate the values associated with the physical world led inevitably to new mathematics which were explicitly suited for that purpose.

The mathematical descriptions of natural laws are called theories, and the key property of these theories is that they can be used to make predictions about the phenomena that they describe. This ability to make predictions is what leads to the belief that the natural laws are immutable and are regular and continuous. We test these theories using experiment. If this were not true, then we would be left with empiricism, which is the merely cataloging of events and values. These events and values may then repeat themselves in the future, but we have no way of knowing this just from the past events. This confidence that we will continue to find laws of nature Wigner calls the "empirical law of epistemology"

Economist seem to have been guided to some extent by the success of physicists in using math to create theories, and have sought to emulate this success in economics. This is a valid course to attempt as the payoff would be so great. But the caution here is that economist must also evaluate the applicability of mathematical descriptions to the laws of economics that they seek to create and be willing to reject those that don't stand up to testing. Perhaps even abandon attempts to use math as the language of economics if it is not appropriate.

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